Show that the points (1, 1), (-1, 5), (7, 9) and (9, 5) taken in that order are the vertices of a rectangle.

Explanation

Let A(1, 1), B (-1, 5), C(7, 9) and D(9, 5) be the vertices of quad. ABCD. Then,
`AB^(2) = (-1-1)^(2) + (5-1)^(2)`
`= (-2)^(2) + 4^(2) = 4+16 = 20.`
`therefore AB = sqrt(20) "units"= sqrt(4 xx 5)` units.
` = 2sqrt(5)` units.
`BC^(2) = (7+1)^(2) + (9-5)^(2)`
`= 8^(2) + 4^(2) = 64 + 16 =80`
`therefore BC = sqrt(80) "units" = sqrt(16 xx 5) "units" = 4sqrt(5)` units.
`CD^(2) = 9-7^(2) + 5-9^(2)`
` =2^(2) + (-4)^(2) = 4 + 16 =20`
`therefore CD = sqrt(20) "units" = sqrt(4 xx 5) "units" = 2sqrt(5) "units. "`
`AD^(2) = (9-1)^(2) + (5-1)^(2)`
` = 8^(2) + 4^(2) = 64 +16 =80`
`therefore AD = sqrt(80) units = sqrt(16 xx 5) "units" = 4sqrt(5)` units.
Thus, `AB =CD = 2sqrt(5) " units and "BC = AD = 4sqrt(5)` units.
Also, `AC^(2) = (7-1)^(2) + (9-1)^(2)`
` = 6^(2) + 8^(2) = 36+64 = 100`
`therefore AC = sqrt(100) "units" = 10 "units"`
`"And,"BD^(2) = (9+1)^(2) + (5-5)^(2) = 10^(2) +0^(2) = 100`
`therefore BD = sqrt(100)` units = 10 units.
`therefore` diagonal AC = diagonal BD.
Thus, ABCD is a quadrilateral whose opposite sides are equal and the diagonals are equal.
Hence, quad. ABCD is a rectangle.
Area of rectangle ABCD `= "length" xx "breadth"`
` = 4sqrt(5) xx 2sqrt(5)` sq units
=40 sq units.